Abstract
Topology optimization has developed rapidly, primarily with application on linear elastic structures subjected to static loadcases. In its basic form, an approximated optimization problem is formulated using analytical or semi-analytical methods to perform the sensitivity analysis. When an explicit finite element method is used to solve contact–impact problems, the sensitivities cannot easily be found. Hence, the engineer is forced to use numerical derivatives or other approaches. Since each finite element simulation of an impact problem may take days of computing time, the sensitivity-based methods are not a useful approach. Therefore, two alternative formulations for topology optimization are investigated in this work. The fundamental approach is to remove elements or, alternatively, change the element thicknesses based on the internal energy density distribution in the model. There is no automatic shift between the two methods within the existing algorithm. Within this formulation, it is possible to treat nonlinear effects, e.g., contact–impact and plasticity. Since no sensitivities are used, the updated design might be a step in the wrong direction for some finite elements. The load paths within the model will change if elements are removed or the element thicknesses are altered. Therefore, care should be taken with this procedure so that small steps are used, i.e., the change of the model should not be too large between two successive iterations and, therefore, the design parameters should not be altered too much. It is shown in this paper that the proposed method for topology optimization of a nonlinear problem gives similar result as a standard topology optimization procedures for the linear elastic case. Furthermore, the proposed procedures allow for topology optimization of nonlinear problems. The major restriction of the method is that responses in the optimization formulation must be coupled to the thickness updating procedure, e.g., constraint on a nodal displacement, acceleration level that is allowed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin Heidelberg New York
Borrvall T (2001) Topology optimization of elastic continua using restriction. Arch Comput Methods Eng 8:351–385
Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190:4911–4928
Bruns TE, Sigmund O (2004) Toward the topology design of mechanisms that exhibit snap-through behavior. Comput Methods Appl Mech Eng 193:3973–4000
Buhl TE, Pedersen CBW, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscipl Optim 19:93–104
Ebisugi T, Fujita H, Watanabe G (1998) Study of optimal structure design method for dynamic nonlinear problem. JSAE Rev 19:251–255
Eschenauer HA, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54:331–390
Fredricson H, Johansen T, Klarbring A, Petersson J (2003) Topology optimization of frame structures with flexible joints. Struct Multidiscipl Optim 25:199–214
Hallquist JO (1998) LS-DYNA theoretical manual. Livermore Software Technology Company, Livermore
Hallquist JO (2002) LS-PRE/POST v1.0. Livermore Software Technology Company, Livermore
Hallquist JO (2004) LS-DYNA user’s manual v. 970. Livermore Software Technology Company, Livermore
Pedersen CBW (2003) Topology optimization design of crushed 2Dframes for desired energy absorption. Struct Multidiscipl Optim 25:368–382
Pedersen CBW (2004) Crashworthiness design of transient frame structures using topology optimization. Comput Methods Appl Mech Eng 193:653–678
Petersson J (1998) Some convergence results in perimeter-controlled topology optimization. Comput Methods Appl Mech Eng 171:123–140
Rainsberger R (2001) TrueGrid Manual v2.1.0 XYZ. Scientific Applications, Livermore
Sigmund O (2001a) Design of multiphysics actuators using topology optimization—Part I: one-material structures. Comput Methods Appl Mech Eng 190:6577–6604
Sigmund O (2001b) Design of multiphysics actuators using topology optimization—Part II: two-material structures. Comput Methods Appl Mech Eng 190:6605–6627
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh dependencies and local minima. Struct Optim 16:68–75
Soto CA (2001a) Optimal structural topology design for energy absorption: a heuristic approach. In: Proceedings of the 2001 ASME design engineering technical conference, Pittsburgh, PA, USA, DETC01/DAC-21126, 9–12 September 2001
Soto CA (2001b) Structural topology optimization: from minimizing compliance to maximizing energy absorption. Int J Veh Des 25:142–163
Soto CA (2002) Applications of structural topology optimization in the automotive industry: past, present and future. In: Fifth world congress on computational mechanics, Vienna, Austria, 7–12 July 2002
Torstenfeldt B (1998) Trinitas developer studio. http://www.solid.ikp.liu.se/~botor/TRINITAS_Learning_Studio.html
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Forsberg, J., Nilsson, L. Topology optimization in crashworthiness design. Struct Multidisc Optim 33, 1–12 (2007). https://doi.org/10.1007/s00158-006-0040-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-006-0040-z